Jan 9 
Jeffrey Galkowski (NEU) 
Eigenfunctions: What are they and what do they tell us?
abstract±
We give an introduction to Laplace eigenfunctions including an overview of some of the important problems from the last century. Laplace eigenfunctions are a useful model problem in a variety of situations and their behavior can be observed in the oscillation of drum membranes, the behavior of electrons, scattering of sound waves, and many other physical systems. Towards the end of the hour, we will give a heuristic discussion of microlocal methods and results.
Poster.

Jan 16 
Alexander Moll (NEU) 
An Introduction to Dispersive Shock Waves
abstract±
At this informal talk, we will define traveling waves and dispersive shock waves of dispersive wave equations, watch experiments and simulations of them on a projector, summarize the heuristics in Whitham modulation theory, ponder what Liouville integrability should mean for wave equations, and contemplate a remarkable singular rational curve from DobrokhotovKrichever (1991).
Poster.

Jan 23 
Changho Han (Harvard) 
Modular compactifications of moduli spaces in algebraic geometry
abstract±
In this talk, we give an introduction to moduli spaces as a way to understand families of objects in consideration. Then, we will see different compactifications of moduli spaces, specially the moduli spaces of curves. If time remains, we will introduce compactifications of moduli of surfaces and their geometry.

Jan 30 
ChiYun Hsu (Harvard) 
Congruences of modular forms and the eigencurve
abstract±
One way to think about a modular form is
through its qexpansion, a power series in
q. Different modular forms can have congruent
qexpansions modulo a prime power, and this
phenomenon inspired Coleman and Mazur to
construct a geometric object called the
eigencurve. In this talk, I will illustrate
congruences of modular forms by examples. I
will also explain how the arithmetic of
modular forms translates into geometry of the
eigencurve.
Poster.

Feb 6 
Peter Hintz (MIT) 
General relativity and microlocal analysis
abstract±
The study of Einstein's field equation in general relativity leads to many interesting and challenging problems in the area of partial differential equations. I will focus on the second simplest type of solution of the field equation, black holes, and describe some questions about their nearequilibrium behavior, and interactions with other black holes. Tools from microlocal analysis and spectral theory are key to recent answers to some of these, and I will sketch how and why.
Poster.

Feb 13 
Jonathan Wang
(MIT) 
Geometric approach to local Lfactors and spherical varieties
abstract±
Recent work suggests that the theory of spherical varieties may help provide a link between period integrals and special values of automorphic Lfunctions. More specifically, the local factor of the Lfunction is related to harmonic analysis on the spherical variety. For certain spherical varieties, these Lfactors were computed at the level of functions by Sakellaridis. However for more general computations, it is expected that a more geometric (in the sense of sheaves replacing functions) approach is needed. I will explain the setup for such an approach and current efforts in this direction.
Poster.

Feb 20 
Tom Bachmann (MIT) 
Affine grassmannians in A^1homotopy theory
abstract±
A^1homotopy theory was invented by Morel and Voevodsky to provide a setting for studying algebraic varieties by homotopical methods. In homotopy theory, important objects of study are *loop spaces*. There is an analog of this in A^1homotopy theory, the socalled Gmloop spaces. Unfortunately Gmloop spaces seem exceptionally hard to study. In this talk I will explain essentially the only known computation of a Gmloop space of a variety: namely, the Gmloop space of a suitable algebraic group coincides with its affine grassmannian.
Poster.

Feb 27 
Eunice Kim (Tufts) 
Mathematical billiards: testing grounds for dynamicists
abstract±
Mathematical billiards is the study of a mass point moving along the geodesics inside a domain and making mirrorlike reflections at the boundary. Billiards first appeared in the context of Boltzmann ergodic hypothesis to study ideal gas, and since then it has served as a useful model for physical systems with collisions. In this talk, through physical examples such as a twoball system and a coin toss model, we will explore how various topics in dynamical systems arise in billiards.
Poster.

Mar 6 
No Seminar. 
Spring Break.

Mar 13 
Euan Spence (U. Bath) 
Why analysts should care about numerical analysis of wave
propagation problems
abstract±
In this talk I will try to convince you that *numerical
analysis* of the PDEs governing wave propagation provides a rich
source of problems for the *analysis* of these PDEs. Note that
absolutely no knowledge of any numerical analysis will be necessary to
understand this talk.
Poster.

Mar 20 
Elina Robeva (MIT) 
Maximum likelihood estimation under total positivity
abstract±
Nonparametric density estimation is a challenging statistical problem  in general the maximum likelihood estimate (MLE) does not even exist! Introducing shape constraints allows a path forward. In this talk I will discuss nonparametric density estimation under total positivity (i.e. logsupermodularity). Though they possess very special structure, totally positive random variables are quite common in real world data and exhibit appealing mathematical properties. Given i.i.d. samples from a totally positive and logconcave distribution, we prove that the MLE exists with probability one if there are at least 3 samples. We characterize the domain of the MLE, and give algorithms to compute it. If the observations are 2dimensional or binary, we show that the logarithm of the MLE is a piecewise linear function and can be computed via a certain convex program. Finally, I will discuss statistical guarantees for the convergence of the MLE, and will conclude with a variety of further research directions.
Poster.

Mar 27 
Lynnelle Ye (Harvard) 
Slopes in eigenvarieties for definite unitary groups
abstract±
The study of eigenvarieties began with Coleman and Mazur, who constructed the first eigencurve, a rigid analytic space whose points are in bijection with padic modular Hecke eigenforms. Since then various authors have constructed eigenvarieties for many other kinds of automorphic forms. We will define automorphic forms on definite unitary groups and explain Chenevier's construction of eigenvarieties for these forms. If time permits, we will state some bounds on the eigenvalues of the U_p Hecke operator appearing in these eigenvarieties.
Poster.

Apr 3 
Geoffrey Smith (Harvard) 
Lowdegree points on curves
abstract±
Faltings' theorem asserts that a smooth curve C of genus at least 2 over a number field K has only finitely many rational points. Given this fact, one can consider the union C(e) of all the sets C(L), where L is any extension of K of degree at most some fixed integer e, and ask whether C(e) is finite. It is easy to see that if C(e) is finite, than e is less than the degree of a map from C to the projective line. This provides an “expected" answer to the question. In this talk, I'll connect this question to a problem in complex algebraic geometry, and, by solving that problem, give several new example families of curves for which the question of finiteness of C(e) has the expected answer. This talk is based on work joint with Isabel Vogt.
Poster.

Apr 10 1:00pm Lake Hall 509 
Scott Mullane (Frankfurt) 
Differentials of the second kind, positivity and
irreducibility of Hurwitz spaces
abstract±
The strata of abelian differentials in the moduli space of pointed curves inform many aspects of the birational geometry of these moduli spaces when g>0. This talk will focus on the strata of differentials of the second kind, special loci inside meromorphic strata that arise as a result of setting all residues to zero and the relation of these strata to the open questions of the Fconjecture and the irreducibility of Hurwitz spaces for specified ramification profile.
Poster.

Apr 17 
Ana Balibanu (Harvard) 
The wonderful compactification
abstract±
Every semisimple complex algebraic group of adjoint type has a canonical smooth equivariant compactification, which fits into a broader context in the world of spherical varieties. I’ll give (time permitting) several constructions of this compactification, and I’ll describe some of its wonderful properties.
Poster.

Apr 24 Canceled 
Monika Pichler (Northeastern) 



End of Semester (Seminar Resumes Fall 2019) 